Opening remarks

• Tetsuo Hatsuda (iTHEMS/Nishina Center)

Room: Integrated Innovation Building

From multi-instantons to exact results

• Jean Zinn-Justin (CEA Saclay)

Room: Integrated Innovation Building We review conjectures about the exact semi-classical expansion of low lying energy levels for analytic potentials with degenerate minima. They take the form of generalized Bohr–Sommerfeld quantization formulae. They were initially motivated by semi-classical evaluations of the partition function based on the path integral formalism (instanton calculus) and have later been proven, to some extent, using the theory of resurgent functions. We explain their relation with the corresponding complex WKB expansion of the solutions of the Schr¨odinger equation, or alternatively of the Fredholm determinant det(H − E). Finally, we recall how these conjectures emerge from a leading order summation of multi-instanton contributions to the path integral representation of the partition function, because the same strategy could result in new conjectures for problems where our present understanding is more limited.

Quantum Geometry and Resurgent Perturbative/Non-perturbative Relations

• Gerald Dunne (U. of Connecticut)

Room: Integrated Innovation Building Certain quantum spectral problems have the remarkable property that the formal perturbative series for the energy spectrum can be used to generate all other terms in the entire trans-series, in a completely constructive manner. I explain a geometric all-orders WKB approach to these perturbative/non-perturbative relations, which reveals surprising connections to number theory and modular forms.

Resurgence Structure to All Orders of Multi-bions in Deformed SUSY Quantum Mechanics

• Toshiaki Fujimori (Keio U.)

Room: Integrated Innovation Building I will discuss resurgence structures in supersymmetric quantum mechanical models with a small SUSY breaking deformation parameter \delta\epsilon. By using the standard Rayleigh-Schroedinger perturbation theory, we exactly determine how the ground state energy respond to the deformation. The full resurgent trans-series expressions for the \delta\epsilon expansion coefficients of the ground state energy are found exactly in CP^1 quantum mechanics. We then discuss the semi-classical bion contributions in the complexified path integral formalism and compare them with the exact results. We determine all exact saddle point solutions corresponding to multi-bion configurations in the complexified theory. By evaluating the complexified quasi-moduli integral, we obtain all order multi-bion contributions which are consistent with the exact results. We also discuss similar resurgence structures in the CP^{N-1} and sine-Gordon models.

Exact WKB analysis of the Gauss hypergeometric differential equation

• Takashi Aoki (Kindai U.)

Room: Integrated Innovation Building The Gauss hyperometric differential equation is investigated from the viewpoint of exact WKB analysis. There are three parameters α, β and γ in the equation. We consider the case where these parameters are linear func- tions of a large parameter. Then the equation can be analyzed by using the standard theory of exact WKB analysis. We can construct WKB solutions of the equation and take the Borel sum of them. Thus obtained analytic solutions can be related to the Gauss hypergeometric function with the large parameter. As an application, asymptotic expansion formulas for the hyper- geometric function in terms of WKB solutions are obtained. These formulas contain asymptotic formulas for the Jacobi polynomials as a special case.

on Voros coefficients in exact WKB analysis

• Tatsuya Koike (Kobe U.)

Room: Integrated Innovation Building A Voros coefficient is one of the most important objects in exact WKB analysis. In this talk, after recalling its definition and background, we review some aspect of a Voros coefficient and its applications: (1) We will see some concrete expressions and their derivation of Voros coefficients for equations of special functions, such as Weber equation, Whittaker equations, and so on. In its derivation, the ladder operators are essentially used. (2) By using the explicit form of Voros coefficients, we can find the analytic structure of the Borel transform of WKB solutions. As its application, we can describe the Stokes phenomenon of WKB solutions when a parameter included in the equation in question varies. (3) To study the analytic structure of the Borel transform of WKB solutions of a general equation, we employ transformation theory. We will briefly recall this.

Iterated convolution and resurgence

• Shingo Kamimoto (Hiroshima U.)

Room: Integrated Innovation Building We introduce the notion of $\Omega$-resurgence associated with a discrete filtered set $\Omega$ following Candelpergher-Nosmas-Pham. We describe the singularity structure of iterated convolution products of such resurgent functions. We further discuss the resurgence of formal series solutions of nonlinear ODE. This talk is partially based on a joint work with David Sauzin.

Resurgent Asymptotics in String Theory

• Ricardo Schiappa (IST, U. of Lisbon)

Room: Integrated Innovation Building I will begin with a light introduction to resurgent asymptotics. These techniques will then be explored (again in the spirit of a light introduction) within the context of transseries solutions to topological and non-critical string theories; themselves obtained via a nonperturbative completion of the holomorphic anomaly equations, or a nonperturbative string equation, respectively.

Hofstadter, Toda and Calabi-Yau

• Yasuyuki Hatsuda (Rikkyo U.)

Room: Integrated Innovation Building The Hofstadter problem is an old topic in condensed matter physics. It describes the spectrum of a Bloch electron in a uniform magnetic field on 2d lattice. In this talk, I will point out that this problem is similar to the eigenvalue problem of a generalized Toda lattice with two particles. It turns out that both problems are related to the Calabi-Yau geometry. I will also discuss the semi-classical analysis in these models.

Nonperturbative ambiguity in double-well type matrix models

• Tsunehide Kuroki (Kagawa College)

Room: Integrated Innovation Building We observe how nonperturbative ambiguities become absent in SUSY/non-SUSY matrix models with double-well type potential, which provide nonperturbative formulations of SUSY/non-SUSY noncritical string theories. We find resurgence structure is quite different in SUSY/non-SUSY correlation functions, even in the same model.

Borel resummation and Exact results in supersymmetric gauge theories

• Masazumi Honda (Weizmann)

Room: Integrated Innovation Building In the past decade, there appeared many exact results in supersymmetric gauge theories thanks to localization method. These exact results are useful to study properties of perturbative series in quantum field theory (QFT) especially in the following two sense. First we can systematically analyze perturbative series around (non-)trivial saddle points in QFT. Second when we manage to obtain resummation of perturbative series without ambiguities in someway (e.g. Borel resummation and resurgence), one can explicitly see how the resummation is related to the exact result including non-perturbative corrections. In my talk I will demonstrate this in 3d N=2, 4d N=2 and 5d N=1 supersymmetric gauge theories on sphere. I will first discuss that we can show Borel summability of perturbative series by Yang-Mills coupling along real positive axis in 4d N=2 and 5d N=1 supersymmetric gauge theories with Lagrangians for various observables. It turns out that exact results in these theories can be obtained by summing over the Borel resummations with every instanton number. I will also discuss perturbative series in general 3d N=2 supersymmetric Chern-Simons matter theory, which is given by a power series expansion of inverse Chern-Simons levels. For this case we can prove that the perturbative series are always Borel summable along imaginary axis while it is not Borel summable along real positive axis often. It turns out that the Borel resummations along this direction are the same as exact results. I will also give physical interpretations of infinite singularities in Borel plane for this class of theories. [PRL116,no.21,211601(2016), PRD94, no.2, 025039 (2016) and upcoming paper(s)]

On confinement in Yang-Mills theory

• Kazuya Yonekura (IPMU, U. of Tokyo)

Room: Integrated Innovation Building I will discuss confinement in Yang-Mills theory in a weak coupling setup. The theory is put on a three-dimensional torus, and the relation between four dimensional Yang-Mills theory and two dimensional CPN model is discussed and used to derive properties of Yang-Mills theory in the confining phase. The vacuum structure of Yang-Mills theory which are usually associated to strong coupling dynamics will be reproduced in completely weakly coupled regime.

Self-resurgence, Cheshire Cat resurgence and the BenderWu algorithm

• Tin Sulejmanpasic (ENS, Paris)

Room: Integrated Innovation Building I will discuss several QM systems and their resurgent properties. With a one-parameter deformation of the double-well and sine-Gordon QM one can obtain systems where part of the spectrum is (almost-)exactly solvable. This feature is intimately related to fact that nonperturbative complex contributions couple to ambiguous complex phases which at special points become non-ambigous. A BenderWu recursive algorithm is used to analyze the perturbation theory of such systems and its relation to non-perturbative physics. We find in this class of systems that special relationships holds between early terms and late terms of the pertrubative series. If time permits I will also discuss the BenderWu algorithm and perturbative behavior of difference equations appearing in the context of topological string theories.

Semiclassics in QCD

• Aleksey Cherman (U. of Washington)

Room: Integrated Innovation Building One big dream in QFT is to use resurgence techniques to study QCD. This requires a controlled semiclassical expansion to exist in the first place, and, for it to be useful, the physics should be smooth in the expansion parameter. I'll explain the current understanding of when such expansions have been worked out, and discuss what we've learned from them. Along the way we will see that QCD observables call for a transseries that involves iterated exponentials of the inverse of the expansion parameter.

PT-symmetry, nonlinear eigenvalue problems and Painleve transcendents

• Carl Bender (Washington U., St.Louis)

Room: Integrated Innovation Building Semiclassical (WKB) techniques are commonly used to find the large-energy behavior of the eigenvalues of linear time-independent Schr\"odinger equations. In this talk we generalize the concept of an eigenvalue problem to nonlinear differential equations. The role of an eigenfunction is now played by a separatrix curve, and the special initial condition that gives rise to the separatrix curve is the eigenvalue. The Painlev\'e transcendents are examples of nonlinear eigenvalue problems, and semiclassical techniques are devised to calculate the behavior of the large eigenvalues. This behavior is found by reducing the Painlev\'e equation to the linear Schr\"odinger equation associated with a non-Hermitian $\cPT$-symmetric Hamiltonian. The concept of a nonlinear eigenvalue problem extends far beyond the Painlev\'e equations to huge classes of nonlinear differential equations.

What is QFT: Resurgence, transseries, and Lefschetz thimbles

• Mithat Unsal (North Carolina State U.)

Room: Integrated Innovation Building I will review a set of connected ideas that provide either a non-perturbative continuum framework or infrared constraints for asymptotically free QFTs. The ideas are large-N volume independence, adiabatic continuity, resurgence and trans-series, Lefschetz thimbles, discrete mixed ’t Hooft anomalies.

Rrenormalization-group Construction of the Invariant/Attractive Manifold of Regular and Stochastic Dynamics

• Teiji Kunihiro (Kyoto U.)

Room: Integrated Innovation Building Some class of dynamical systems can be reduced to a slower dynamical system with fewer variables, which usually forms an invariant or attractive manifold in the asymptotic regime.We show that the perturbative renormalization-group (RG) method can be nicely formulated as a natural and simple method for deriving the slow dynamics and the invariant manifold on which the slow dynamics is defined: We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation has no Jordan cell; when A has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory. We present some examples for the RG reduction and construction of the invariant manifold, which may include a derivation of fluiddynamical equations from Boltzmann equation, Fokker-Planck equation from Langevin equation, a further adiabatic elimination of fast variables in the Fokker-Planck equation.

Reconstruction of the tunneling amplitude from the perturbation series

• Hiroshi Suzuki (Kyushu U.)

Room: Integrated Innovation Building I give a retrospective review on how we arrived at a formula which extracts the tunneling rate from a non Borel summable perturbation series in a quantum mechanical example. The formula gives rise to a very accurate tunneling rate, especially in the strong coupling regime. The keywords are: the (scaled) delta expansion, the Borel singularity, and the conformal mapping on the Borel plane.

An old story: From the valley instantons to a sypersymmetric Q.M

• Hideaki Aoyama (Kyoto U.)

Room: Integrated Innovation Building I wil present our works in the 90's on interrelations between non-perturbative and perturbative effects in quantum theories. We introduced the `new` valley method, which allowed us to define multi instanton and anti-instanton configurations. This enabled us to beyond dilute gas approximation in Q.M. and Q.F.T.. Working on Q.M. with an asymmetric double well potential in 1 dim. space, we constructed valley instantons (bounces) both numerically and analytically, by using which we then revealed the inter-relation between the perturbatice and the nonperturbative contributions, first proposed by Bogomolny. Exact numerical calculation of perturbative series up to several hundred orders then lead to the dicovery of a series of new polynomical supersymmetry, "N-fold supersymmetry", in this system.

Theta dependence and anomaly matching

• Yuya Tanizaki (RIKEN BNL)

Room: Integrated Innovation Building I will reconsider about the semiclassical treatment of the theta angle dependence of quantum physics. Using a quantum mechanical model, I want to emphasize the importance of twisted boundary condition, and I will investigate its connection to recent developments of 't Hooft anomaly and global inconsistency.

Closing remarks Room: Integrated Innovation Building